Visualizing Multi-D Objects (IMUC 2009)

Single-Variable calculus is commonly taught from a graphic standpoint, using the concepts of area and tangent lines to describe the world of integrals and derivatives. However, when we move to multivariable calculus, most students ask the question: what does a 4D surface look like? Without this idea, it is difficult to draw the connection between techniques learned in elementary calculus and techniques in the 4th dimension and beyond. We certainly cannot pull out a hypersphere from our textbooks. Two methods of visualization using Mathematica are proposed.It is plausible to describe a 3D object using 2D images by plotting the “shadows” that are cast on 2D planes (subspaces). We can use the same idea to visualize a higher-dimension object by plotting its projections on a systematic set of 3D subspaces. In order to do this, a projection matrix can be making a spanning matrix of the target 3D subspace, then multiplying it by its pseudoinverse. Hence, we can do a linear transformation of any multi-dimensional object into a plottable 3D surface.

Technique 1: Projecting on a Systematic Set of Coordinate Axes. This method generates a list of 3-D subspaces for the higher dimensions by coming up with all the different combinations of coordinate axes of the higher dimension, by taking 3 at a time. This is more intuitive for us. Visualizing a 5D hypersphere is presented.

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Technique 2: Projecting onto the Null Space of a Selected Point. The second one allows the user to choose a point on a 4D surface. The surface is then projected onto the null space (aka perpendicular space) of that point. The null space of a point is defined as the set of all vectors that will linearly transform the point into the null vector. Visualizing a 4D hypersphere is presented.

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Afterthoughts:

When we take the 3D projection of a hypersphere, we do not always get spheres. On the other hand, if we have a 3D sphere and project it on 2D planes, we always get a circle. However, when we project on the subspace consisting of the xyz-axes, we do end up with the original unit sphere. Also, projecting a hypersphere on the null space of a point on a sphere produces a sphere-like object that collapses on itself becasue of the nature of the null space.

Although this is only a crude approximation of 4D objects, these techniques can help wrap our mind around such an abstract idea. This shows that the various dimensions can be connected to each other somehow, and are not as foreign as assumed.